The first two operators in each column exist in both intuitionistic and dual-intuitionistic propositional logic and the last two in each column exist in both forms of predicate logic and modal logic (respectively), but they are still dual as shown. All of these exist in classical logic (although some of the paraconsistent operators are not widely used), and the two forms of negation (¬\neg and −-) are the same there.

In linear logic, this extends to a duality between conjunctive and disjunctive operators:

since every other aspect of the first two lines is already constructively valid, the claim that negation mediates the de Morgan self-duality of negation already has a name (the double negation law, equivalent to the principle of excluded middle), and no other line involves only operators that appear in intuitionstic propositional calculus.